340 The Astronomer Royal's Remarks on Prof. Challis's 



mathematical investigation. On page 278 a constant — i* 

 {whose sign is fixed without any ostensible reason, but for 

 which I shall hereafter give a reason} has been introduced; 

 it is indeterminate ; there is no a priori reason for thinking 

 that it has any assignable value, or that it is needed at all: 

 but as it is possible that such a constant may be admissible, 

 it is quite proper that it should be introduced for trial in the 

 subsequent parts of the investigation. It is accordingly intro- 

 duced and tried, and we immediately perceive that, if it has a 

 value different from zero, the result of non-diffusion of vibra- 

 tions, which Professor Challis considers tacitly as the first 

 object to be secured, (see p. 281, line 10 from the bottom,) 

 cannot be obtained. Most reasoners would conclude, either 

 that such a result is not legitimately to be expected, or that a 

 last trial should be made by supposing 6^ = 0. Instead of this. 

 Professor Challis has recourse to another equation, extracted 

 from another memoir, and not demonstrated here, which he 

 considers more accurate than his equation ('!?.) ; and he then 

 uses this new equation in conjunction with equation (3.), and 

 thus obtains the startling results to which I have alluded. 



I must request you to record my protest against the intro- 

 duction of this new equation. The equations (3.) and (4.) 

 have been fairly and iionestly obtained together, and they 

 ought to be fairly and honestly retained together. One cannot 

 be obtained without the other, and one ought not to be changed 

 without reinvestigating the other and changing it if necessary. 



I must also object to the application of the word "approxi- 

 mate" to the equation (i.). The equation is perfectly accu- 

 rate as far as the first order of disturbances of particles, which 

 is the limit of the investigations in this paper. 



As the remainder of the paper, except the last paragraph, 

 is entirely founded upon the introduction of the new equation, 

 I conceive that it offers no evidence whatever for Professor 

 Challis's conclusions. 



I will now point out what I conceive to be the legitimate 

 conclusions from equations (3.) and (4.). 



(«.) We may suppose ^ = or ^ = 0. The equation (6.) is 

 then reduced to 



^+1.^ = 

 dr'^^ r dr ' 



and the equation (4.) is reduced to 



^ + ^ =0. 

 dx^ dy^ 



The general solution of the former of these equations con- 



